Dear friends.
Is it obligatory enphasizing by inversion in this sentence?:
"No sooner did I enter the room than the door slamed."
Or I can say: "No sooner I entered the room than the door slamed."
What about this other example:" So nice the day was that we went out"
Is it correct or I should say:"So nice was the day..."
Is this way of emphasizing common in speaking or just in writing?
Thank you.

You can say, "As soon as I entered the room the door slammed." That would be more usual.
"No sooner I entered the room than the door slamed." Is incorrect, it needs to be written, "No sooner HAD I entered the room than the door slammed."
" So nice the day was that we went out" Is not correct. If you insist on the inversion it would be, "So nice was the day, that we went out."
You would usually hear people say," The day was so nice that we went out" (to the beach etc.,)

I think the kind of inversion we are discussing is used more in writing. Most people don't talk to each other like that. It sounds "affected".